We intuitively say that a variable is a function of a second variable when its value depends on the value of the second variable and the value of the first variable can be uniquely calculated by some rule when the value of the second variable is assumed. The first variable is called the dependent variable and the second the independent variable.

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Notation

In mathematics, we often wish to refer to a generic function without specifying any particular formula, table, or graph. To denote that y is a function of x, we write
y=f(x), which is read as “y is equal to f of x.” In this notation,f represents the function, that is, the “rule” or “procedure” (which usually but not always involves some formula) associating the values of x to the values of y. Instead of f(x), we may use other notations such as g(x),ϕ(x),F(x),f(x),s(x), etc. If more than a function occur in a problem, one may be expressed as f(x), another as F(x), another as g(x), and so on. It is also convenient in practice to represent different functions by the symbols f1(x),f2(x),f3(x), etc. However, during any investigation, the same functional symbol always indicates the same law of dependence of the dependent variable upon the independent variable.

 

Visualizations

A function can be thought of as a machine or a computer program that assigns one output to every allowable input.

(a)  (b)

Figure 1: A function can be thought of as a (a) machine or (b) computer program that for each allowable input gives one output.

Another way to picture a function is by an arrow diagram. For each element x in A, the value of f(x) in B is to be found at the head of the corresponding arrow. As we can see in Figure 2(a), f associates to each element in A, one and only one element in B. Thus, f is a function, although two elements in A are associated with one element in B. But in Figure 2(b), g associates two elements to an element  in A. Therefore, g is not a function.

(a)  (b)

Figure 2: f is a function, but g is not because g assigns two elements in its codomain to in its domain.

 

Definitions

Definition: A function f from a set A, to a set B, is a rule that assigns, to each element x in A, one
and only one element y in B. We then write y=f(x).

Sets A and B are called domain and co-domain of f, respectively. To mention that f is a function with domain A and co-domain B, we write f:AB.

  • If y=f(x), we also call the independent variable x, the argument of the function, and the element y the value of f at x or the image of x under f. 
  • The definition of a function f:AB does not restrict the nature of the elements of A and B, but in elementary calculus we assume that they are numbers; that is, A and B are subsets of real numbers R, unless otherwise stated.

 

Formula of the Function

If y=f(x), the particular value of the function when x has a definite value a is then expressed as f(a). For example, if
f(x)=4x25x+1, then
f(1)=4(1)25(1)+1=10, and
f(0)=4(0)25(0)+1=1. Also, because it does not matter which letter we use for the independent variable, we have
f(t)=4t25t+1, f(u)=4u25u+1, f(b+1)=4(b+1)25(b+1)+1=4b2+3b.

Strictly speaking f(x) is the value of f at x, but we often talk about “the function f(x)” or “the function y=f(x).” For example, consider a function f:RR that takes a number x and gives its square x2. In this case, we can simply say:

1.“the function f(x)=x2;”
2.“the function y=x2” (if we denote the dependent variable by y);
3. simply “the function x2.”

We can also connect the input and output values by a special arrow, namely “the function xx2.”

 

Implicit and explicit functions

If an equation between several variables is solved for any one, the latter is said to be an explicit function of the others, the manner of its dependence being exhibited by the solution of the equation. Otherwise, it is said to be an implicit function. Thus, in x2+y2=4, y is an implicit function of x; while, in y=4x2, y is an explicit function of x. The difference is one of form only. The notation y=f(x) is used to denote that y is an explicit function of x, and the notation f(x,y)=0 to denote that x and y are implicit functions of each other.